Abstract
We compute the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of Fano 3folds. This is the first step in understanding the nontrivial Gerstenhaber algebra structure of this invariant, and yields some initial insights in the classification of Poisson structures on Fano 3folds of higher Picard rank.
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1 Introduction
In this paper we describe the Hochschild cohomology of Fano 3folds, with the eventual goal of understanding the interesting algebraic structures present on this invariant, and completing the classification of Poisson structures on Fano 3folds.
Fano 3folds and the vector bundle method. Fano 3folds were classified by Iskovskikh [24, 25] (for Picard rank 1, where there are 17 families) and Mori–Mukai [40, 41] (for Picard rank \(\ge 2\), where there are 88 families). This classification was obtained by understanding the birational geometry of Fano 3folds, and the output is a list of 105 deformation families and their numerical invariants \(\textrm{c}_1(X)^3\), \(\rho (X)\), and \({{\,\textrm{h}\,}}^{1,2}(X)\). Only 12 out of 88 families of Picard rank \(\ge 2\) are not the blowup of a Fano 3fold of lower Picard rank.
For the Picard rank 1 case Mukai alternatively described the classification using the vector bundle method in [43], by writing Fano 3folds of Picard rank 1 as zero loci of vector bundles on homogeneous varieties and weighted projective spaces. In higher Picard ranks this was extended in 2 different ways (which have partial overlap), by giving a description as

(1)
zero loci of vector bundles on GIT quotients by products of general linear groups [15]; or

(2)
zero loci of homogeneous vector bundles on homogeneous varieties [16].
The ambient variety is often called the key variety, and will be denoted F.
In the first variation on the vector bundle method the group is often (but not always) a product of tori, so that the Fano 3fold is described as a complete intersection in a toric variety F. There are 13 families for which the group is not a product of tori, and then the key variety is actually a product of Grassmannians.
In the second variation the key variety F is always a homogeneous variety. In particular, it is shown in [16] that every Fano 3fold can be realised as the zero locus of a homogeneous vector bundle in a product of (possibly weighted) Grassmannians.
Hochschild cohomology. We will use both these descriptions to determine the Hochschild cohomology of all Fano 3folds. This is an invariant which measures the deformation theory of the abelian category (or derived category) of (quasi)coherent sheaves [36, 37, 54]. For the definition and more details on the algebraic structure on \({{\,\textrm{HH}\,}}^\bullet (X)\), see Sect. 2.1.
An important instrument in describing Hochschild cohomology for varieties is the Hochschild–Kostant–Rosenberg decomposition [12, 53, 56], which says that
The birational description of Mori–Mukai is not convenient for automating computations of the righthand side of (1) for Fano 3folds, whereas the vector bundle method turns out to be wellsuited for this. Moreover, we need the combination of both descriptions to cover all Fano 3folds, together with a separate analysis of some underdetermined cases, i.e. cases for which the two approaches do not yield a complete description of the cohomology groups we are aiming at.
In Sect. 3.1 we will show how to compute \({{\,\textrm{h}\,}}^p(X,\bigwedge \nolimits ^q\textrm{T}_X)\) for \(q \ne 2\): the summands with \(q=0\) and \(q=3\) are easy, and the summands for \(q=1\) follow from the knowledge of the invariants \(\textrm{c}_1^3\), \(\rho \), and \(\textrm{h}^{1,2}\) together with the size of the automorphism group of a Fano 3fold X. For \(q=2\) the description is new, and forms the main subject of this paper.
Theorem A
Let X be a Fano 3fold. Then the cohomology of \(\bigwedge \nolimits ^2\textrm{T}_X\) is concentrated in degrees 0, 1, 2, and it is constant in families. The dimensions of the cohomologies (for all \(q=0,1,2,3\)) are given in the tables in Appendix A.
The fact that the dimension of cohomology is constant in families is a byproduct of the calculations, we don’t have an abstract proof for it. Observe that the cohomology of the tangent bundle is not constant in families, see [47] for the jumping behaviour of \({{\,\textrm{Aut}\,}}^0(X)\) and therefore \({{\,\textrm{h}\,}}^0(X,\textrm{T}_X)=\dim {{\,\textrm{Aut}\,}}^0(X)\).
On the methods. In Sects. 3 and 4 we collect the details for the proof of Theorem A. We will set up the proof so that we can take advantage of computer algebra methods, with some explicit calculations in cases where automated methods fail. We have optimised the automated methods so that only 5 out of 105 deformation families of Fano 3folds need to be dealt with by hand (2 of which are nearly immediate).
What is interesting to observe is that the homogeneous methods from [16] are very good at determining Hodge numbers (and in particular they are expected to help in classifying Fano 4folds, see e.g. [8]), with only a dozen deformation families of Fano 3folds not being fully determined. But for twisted Hodge numbers (and in particular the cohomology of \(\textrm{T}_X\) and \(\bigwedge \nolimits ^2\textrm{T}_X\)) the homogeneous approach gives many underdetermined cases.
This is why we first use the toric description from [15], and only use the homogeneous description when no such description is available or when the toric methods are not giving a full answer. The combination of these two methods, in this particular order, gives the cleanest exposition.
Absence of Poisson structures. When \({{\,\textrm{H}\,}}^0(X,\bigwedge \nolimits ^2\textrm{T}_X)\) is nonzero, the classification of Poisson structures becomes an interesting question. For a global bivector, the vanishing of the selfbracket (for the Schouten bracket) is equivalent to the Jacobi identity of the associated Poisson structure.
In [35, §9, Table 1] Poisson structures on Fano 3folds of Picard rank 1 were classified (see also [2] for the classification of Poisson structures on smooth projective surfaces, where the vanishing is automatic). As an immediate corollary of Theorem A we obtain the absence of Poisson structures on some Fano 3folds of higher Picard rank. Here the notation \({{\textrm{MM}}}_{\rho .n}\) refers to the nth deformation family with Picard rank \(\rho \) in the Mori–Mukai classification, see [40] and [26, §12.2\(\)12.6].
Corollary B
The following primitive^{Footnote 1} Fano 3folds with \(\rho \ge 2\) admit no Poisson structures:

\({{\textrm{MM}}}_{2.2}\)

\({{\textrm{MM}}}_{2.6}\)

\({{\textrm{MM}}}_{3.1}\).
The following imprimitive Fano 3folds admit no Poisson structures:

\({{\textrm{MM}}}_{2.4}\)

\({{\textrm{MM}}}_{2.7}\)

\({{\textrm{MM}}}_{3.3}\).
For all other Fano 3folds there are nonzero global bivectors, and it is necessary to check the selfbracket of a global bivector field. Already for Fano 3folds of Picard rank 1 this is a highly nontrivial condition [35].
For the imprimitive Fano 3folds we expect that the birational description of Mori–Mukai together with [46, §8] should allow for a (partial) classification of Poisson structures. In particular, we expect that the second part of Corollary B has a proof using these techniques, but this is outside the scope of the current paper.
Relation to other works. In the representation theory of finitedimensional algebras the Gerstenhaber algebra structure on Hochschild cohomology is an important invariant, studied in many cases, see [1, 14, 49, 51] to name a few. In algebraic geometry there are (at the time of writing) fewer attempts at giving explicit descriptions of Hochschild cohomology and the Hochschild–Kostant–Rosenberg decomposition. An important case is that of partial flag varieties [7, 21]. For smooth projective toric varieties (and only the \({{\,\textrm{H}\,}}^0\), not any possible \({{\,\textrm{H}\,}}^{\ge 1}\)) one is referred to [22]. There are also various cases where the interaction of the Hochschild cohomology of different varieties (and categories) is studied (see e.g. [6, 23, 32]), with the Kuznetsov components of Fano 3folds of Picard rank 1 and index 2 being the subject of [33, §8.3].
Some of the results in this paper are standard, whilst for Fano 3folds of Picard rank 1 results can be found in [27, 35].
It would be interesting to understand how mirror symmetry can be used to compute the invariants investigated in this paper, using the symplectic geometry of the mirror Landau–Ginzburg model. For Hodge numbers (and hence Hochschild homology, see Sect. 2.1) of Fano varieties a recipe for this was conjectured by Katzarkov–Kontsevich–Pantev in [28, Conjecture 3.7], based on the conjectural equivalence
from homological mirror symmetry. Here \(f:Y\rightarrow {\mathbb {A}}^1\) is a (suitably compactified) Landau–Ginzburg model and \(\omega _Y\) an appropriately chosen symplectic form, so that X and (Y, f) are mirror. Subsequently this was checked by Lunts–Przyjalkowski for del Pezzo surfaces in [38] and by Cheltsov–Przyjalkowski for Fano 3folds in [13]. Hochschild cohomology is also a categorical invariant, and therefore can be computed from either side of (2) (assuming an enhancement of the equivalence). An interesting difference is that Hodge numbers (and hence the dimensions of the Hochschild homology spaces) are constant in families, but this is not the case for Hochschild cohomology.
Notation. We will number deformation families of Fano 3folds as \({{\textrm{MM}}}_{\rho .n}\) as in Mori–Mukai [40] (see also [26, §12.2\(\)12.6]), with the caveat that \({{\textrm{MM}}}_{4.13}\) refers to the blowup of \({\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1\) in a curve of degree (1, 1, 3), the case which was originally omitted and discovered in [42].
Throughout we work over \({\mathbb {C}}\), so that we can apply the descriptions of [15, 16] for Fano 3folds. The calculations using these descriptions are in fact valid over an arbitrary algebraically closed field of characteristic zero.
2 Polyector fields and their structure
2.1 Hochschild cohomology and the Hochschild–Kostant–Rosenberg decomposition
There exist various approaches to defining the Hochschild cohomology of a variety, which are known to agree in the setting we are interested in. One of the more economical definitions is the following.
Definition 2.1
Let X be a smooth and projective variety. Its Hochschild cohomology is
for
where \(\Delta :X\hookrightarrow X\times X\) denotes the diagonal embedding.
The Hochschild–Kostant–Rosenberg decomposition gives a convenient description of the summands \({{\,\textrm{HH}\,}}^i(X)\) in terms of polyvector fields, and it is obtained via the Hochschild–Kostant–Rosenberg quasiisomorphism \({\textbf{L}}\Delta ^*\circ \Delta _*{\mathcal {O}}_X\cong \bigoplus _{i=0}^{\dim X}\Omega _X^i[i]\) considered in [12, 39, 56].
Theorem 2.2
(Hochschild–Kostant–Rosenberg decomposition) Let X be a smooth projective variety. Then there exists an isomorphism
for \(i=0,\ldots ,2\dim X\) induced by the Hochschild–Kostant–Rosenberg quasiisomorphism.
Hence as a first approximation (disregarding any algebraic structures present on Hochschild cohomology) determining the Hochschild cohomology of a variety reduces to a question in sheaf cohomology.
Remark 2.3
There is also the Hochschild homology of X, defined as
where
Moreover there is the Hochschild–Kostant–Rosenberg decomposition for Hochschild homology, which now reads
for \(i=\dim X,\ldots ,\dim X\). Hence the dimension of the Hochschild homology of X is determined by the Hodge numbers \({{\,\textrm{h}\,}}^{p,q}={{\,\textrm{h}\,}}^q(X,\Omega _X^p)\). These numbers admit symmetries under Serre duality and Hodge symmetry, and therefore are often written down in the form of the Hodge diamond. In particular for Fano 3folds the Hodge diamond is of the form
and it is determined by the invariants from the classification. The dimensions of the Hochschild homology spaces now correspond to different columns in this diamond (as opposed to the rows which describe the dimensions of singular cohomology spaces).
To mimic this economical description of the Hochschild–Kostant–Rosenberg decomposition of Hochschild homology using the Hodge diamond, the first author introduced the polyvector parallelogram. If we denote \({{\,\textrm{p}\,}}^{p,q}:=\dim _k{{\,\textrm{H}\,}}^p(X,\bigwedge \nolimits ^q\textrm{T}_X)\), then for a 3fold it is given by
with an obvious generalisation to other dimensions. There are no symmetries present in the numbers \({{\,\textrm{p}\,}}^{p,q}\), and the presentation reflects this absence.
Remark 2.4
Another important difference between the Hodge diamond and the polyvector parallelogram is that the former is constant in families, whilst the latter is not necessarily so. We will explain this for Fano 3folds in Sect. 3.1.
Additional structure. There is a rich algebraic structure on Hochschild cohomology \({{\,\textrm{HH}\,}}^\bullet (X)\), and on the polyvector fields \(\bigoplus _{p+q=\bullet }{{\,\textrm{H}\,}}^p(X,\bigwedge \nolimits ^q\textrm{T}_X)\). Namely there exist:

a gradedcommutative product (of degree 0);

a graded Lie bracket (of degree \(1\))
which are related via the Poisson identity, yielding the structure of a Gerstenhaber algebra.
On Hochschild cohomology this structure can be either induced using a localised version of the Hochschild cochain complex of an algebra [30, 56], or the general machinery of Hochschild cohomology for dg categories [29]. The product corresponds to the Yoneda product on selfextensions in (4), whilst the Gerstenhaber bracket \([,]\) does not have a direct sheaftheoretic interpretation in the definition (4).
For polyvector fields the product structure is given by the cup product in sheaf cohomology together with the wedge product of polyvector fields, whilst the Lie bracket is given by the Schouten bracket \([,]_{\textrm{S}}\). In this case the Gerstenhaber algebra structure is even compatible with the bigrading.
The isomorphism used in Theorem 2.2 is not compatible with the Gerstenhaber algebra structures on both sides. This was remedied by Kontsevich (see [31, Claim 8.4] and [12, Theorem 5.1]) for the algebra structure and Calaque–Van den Bergh [11, Corollary 1.5] for the full Gerstenhaber algebra structure, by modifying it using the square root of the Todd class. We will denote the isomorphism \({{\,\textrm{HH}\,}}^\bullet (X)\cong \bigoplus _{p+q=\bullet }{{\,\textrm{H}\,}}^p(X,\bigwedge \nolimits ^q\textrm{T}_X)\) of graded vector spaces obtained from Theorem 2.2 by \(\textrm{I}^{\textrm{HKR}}\).
Theorem 2.5
(Kontsevich, Calaque–Van den Bergh) We have an isomorphism of Gerstenhaber algebras
By describing the algebraic structure on polyvector fields we can therefore deduce properties of the algebraic structure on Hochschild cohomology of X.
We can identify certain interesting substructures:

\(({{\,\textrm{HH}\,}}^1(X),[,])\) is a Lie algebra,

\({{\,\textrm{HH}\,}}^i(X)\) is a representation of \(({{\,\textrm{HH}\,}}^1(X),[,])\),

the selfbracket \([\alpha ,\alpha ]\in {{\,\textrm{HH}\,}}^3(X)\) for \(\alpha \in {{\,\textrm{HH}\,}}^2(X)\) measures the obstruction to extending a firstorder deformation of the abelian or derived category of coherent sheaves (classified by \({{\,\textrm{HH}\,}}^2(X)\), see [36, 37]) to higher order,
whilst on the polyvector fields and using the finer bigrading we have that:

\(({{\,\textrm{H}\,}}^0(X,\textrm{T}_X),[,]_{\textrm{S}})\) is the Lie algebra \({{\,\textrm{Lie}\,}}{{\,\textrm{Aut}\,}}^0(X)\);

\(\bigoplus _{p+q=i}{{\,\textrm{H}\,}}^p(X,\bigwedge \nolimits ^q\textrm{T}_X)\) is a bigraded representation of \({{\,\textrm{Lie}\,}}{{\,\textrm{Aut}\,}}^0(X)\);

the selfbracket \([\beta ,\beta ]_{\textrm{S}}\in {{\,\textrm{H}\,}}^2(X,\textrm{T}_X)\) for \(\beta \in {{\,\textrm{H}\,}}^1(X,\textrm{T}_X)\) measures the obstruction to extending a firstorder deformation of the variety X to higher order in the Kodaira–Spencer deformation theory of varieties.
For a Fano variety the latter obstruction vanishes as \({{\,\textrm{H}\,}}^2(X,\textrm{T}_X)=0\) by Kodaira–Akizuki–Nakano vanishing, see also Lemma 3.1. By [47] the Lie algebra \({{\,\textrm{Lie}\,}}{{\,\textrm{Aut}\,}}^0(X)\) is nontrivial in many cases, and it would be interesting (but outside the scope of this article) to describe this aspect of the Gerstenhaber algebra structure.
There is also the selfbracket \([\pi ,\pi ]_{\textrm{S}}\in {{\,\textrm{H}\,}}^0(X,\bigwedge \nolimits ^3\textrm{T}_X)\) for \(\pi \in {{\,\textrm{H}\,}}^0(X,\bigwedge \nolimits ^2\textrm{T}_X)\), which we will now elaborate on. By Kodaira vanishing \({{\,\textrm{H}\,}}^2(X,{\mathcal {O}}_X)\) will play no role in this article.
2.2 Poisson structures
A Poisson structure is a \({\mathbb {C}}\)bilinear operation \(\{,\}:{\mathcal {O}}_X\times {\mathcal {O}}_X\rightarrow {\mathcal {O}}_X\) satisfying the axioms of a Poisson bracket; in particular it satisfies the Jacobi identity. It can also be encoded globally as a section \(\pi \in {{\,\textrm{H}\,}}^0(X,\bigwedge \nolimits ^2\textrm{T}_X)\), using the equality \(\{f,g\}=\langle \textrm{d}f\wedge \textrm{d}g,\pi \rangle \) obtained from the pairing between vector fields and differential forms. The vanishing of the Schouten bracket
encodes the Jacobi identity for the corresponding Poisson structure. We will use the following terminology.
Definition 2.6
Let X be a smooth projective variety. A Poisson structure on X is a bivector field \(\pi \in {{\,\textrm{H}\,}}^0(X,\bigwedge \nolimits ^2\textrm{T}_X)\) such that (12) holds. We denote
the subvariety of Poisson structures.
In general \({{\,\textrm{Pois}\,}}(X)\) is cut out by homogeneous equations of degree 2, and one can also consider them up to rescaling, so that one is interested in \({\mathbb {P}}({{\,\textrm{Pois}\,}}(X))\subseteq {\mathbb {P}}({{\,\textrm{H}\,}}^0(X,\bigwedge \nolimits ^2\textrm{T}_X))\). There can be multiple irreducible components, of varying dimension. For an excellent introduction to Poisson structures, one is referred to [48]. Let us just recall that Poisson structures are important to construct deformation quantisations, or noncommutative deformations, as e.g. explained in [9].
The classification of Poisson structures on smooth projective surfaces is done in [2], with the vanishing of the Schouten bracket being automatic for dimension reasons. The classification of Poisson structures Fano 3folds of Picard rank 1 is summarised in [35, §9, Table 1]. We don’t need the full classification, let us just mention the following examples.
Example 2.7
By [35, §9, Table 1] we have that

for \({\mathbb {P}}^3\) there are 6 irreducible components, of varying dimension;

in the family \({{\textrm{MM}}}_{1.10}\) there exists a unique member for which \({\mathbb {P}}({{\,\textrm{Pois}\,}}(X))\) is nonempty in \({\mathbb {P}}({{\,\textrm{H}\,}}^0(X,\bigwedge \nolimits ^2\textrm{T}_X))\cong {\mathbb {P}}^2\), in which case it is a point: the Mukai–Umemura 3fold \(X^{\textrm{MU}}\) for which \({{\,\textrm{Aut}\,}}^0(X^{\textrm{MU}})={{\,\textrm{PGL}\,}}_2\);

in the family \({{\textrm{MM}}}_{1.9}\) we have for all X that \({\mathbb {P}}({{\,\textrm{Pois}\,}}(X))=\emptyset \) inside \({\mathbb {P}}({{\,\textrm{H}\,}}^0(X,\bigwedge \nolimits ^2\textrm{T}_X))=\textrm{pt}\).
As mentioned in [48, §3.4], the full classification of Poisson structures on Fano 3folds of higher Picard rank is still open, and Corollary B gives the first step towards such a classification.
3 Computing the Hochschild cohomology of Fano 3folds
In this section we discuss the aspects of the computation of Hochschild cohomology of Fano 3folds which are common to all cases. After introducing some general results in Sect. 3.1 we will set up the computation in Sect. 3.2 and discuss the two approaches in Sects. 3.3 and 3.4. For the remaining cases one is referred to Sect. 4.
3.1 General results
The following lemma is straightforward, but significantly reduces the number of cohomologies one needs to compute for a Fano 3fold.
Lemma 3.1
Let X be a Fano 3fold. Then
Proof
This is immediate from the Kodaira–Akizuki–Nakano vanishing
for an ample line bundle \({\mathcal {L}}\), by considering \((p,{\mathcal {L}})=(3,\omega _X^\vee ),(2,\omega _X^\vee ),(1,\omega _X^\vee ),(3,\omega _X^\vee \otimes \omega _X^\vee )\) and using the identification \(\bigwedge \nolimits ^i\textrm{T}_X\cong \omega _X^\vee \otimes \Omega _X^{3i}\). \(\square \)
In particular, the polyvector parallelogram introduced in Sect. 2.1 has the form
Next we describe the Euler characteristic of the vector bundles appearing in Lemma 3.1. Recall that Hirzebruch–Riemann–Roch for a vector bundle \({\mathcal {E}}\) on a 3fold takes on the following form, where we abbreviate \(\textrm{c}_i=\textrm{c}_i(\textrm{T}_X)\):
We obtain the following identities, expressing the Euler characteristic of the bundles we are interested in in terms of the usual invariants \(\rho \), \({{\,\textrm{h}\,}}^{1,2}\) and \(\textrm{c}_1^3\) in the classification of Fano 3folds.
Lemma 3.2
Let X be a Fano 3fold.
Proof
By (17) for \({\mathcal {O}}_X\) and Kodaira vanishing we have that \(\chi (X,{\mathcal {O}}_X)=\frac{\textrm{c}_1\textrm{c}_2}{24}=1\), so
And \(\textrm{c}_3\) is the topological Euler characteristic, so
Hence (18) and (20) follow from (17).
For (19) we use that
so that reading off the degree three part of \({{\,\textrm{ch}\,}}(\bigwedge \nolimits ^2\textrm{T}_X)\textrm{td}_X\) gives
and the identity in (19) follows from the observations made in the previous paragraph. \(\square \)
This observation, together with the classification of infinite automorphism groups of Fano 3folds (see [34, Theorem 1.1.2] for Picard rank 1, and [47, Theorem 1.2] for Picard rank \(\ge 2\)), makes it straightforward to determine \({{\,\textrm{h}\,}}^0(X,\textrm{T}_X)\) and \({{\,\textrm{h}\,}}^1(X,\textrm{T}_X)\).
Proposition 3.3
Let X be a Fano 3fold. We have that
The computation of \({{\,\textrm{Aut}\,}}^0(X)\) can be found in [47, Table 1]. It is important to note that the dimension of \({{\,\textrm{Aut}\,}}^0(X)\) can vary in families.
For \(\bigwedge \nolimits ^2\textrm{T}_X\) we need to determine 3 possibly nonzero cohomologies, and none is known a priori. Some cases are easy (e.g. for toric Fano 3folds Bott–Steenbrink–Danilov vanishing, see e.g. [45, Theorem 2.4], yields that \({{\,\textrm{H}\,}}^{\ge 1}(X,\bigwedge \nolimits ^i\textrm{T}_X)=0\)) but others take more effort.
3.2 Setting up the computation
As discussed in the previous section, it suffices to compute the cohomology of \(\bigwedge \nolimits ^2\textrm{T}_X\) to fully determine the Hochschild cohomology of a Fano 3fold. By Lemma 3.1 we know that its cohomology is concentrated in degrees 0, 1, 2.
To perform this computation we will use suitable descriptions of Fano 3folds X inside key varieties F provided in [15, 16]. A key variety will be either a product of Grassmannians or a toric variety. In the former case X is given as the zero locus of a general global section of a homogeneous vector bundle \({\mathcal {E}}\) on F. In the latter case X is given as an intersection of divisors inside a possibly singular F. It turns out that this second description involves nonCartier divisors only for \({{\textrm{MM}}}_{2.1}\) and \({{\textrm{MM}}}_{2.3}\): this will lead us to deal with these two cases separately in Sect. 4.
The two methods outlined in this section allow for a near uniform treatment using computer algebra methods. We implemented them using Macaulay2 [20] and Magma [10]; our code is publicly available at [5] and can be used to check our computations. As it turns out, this automated treatment leaves the cohomology of \(\bigwedge \nolimits ^2\textrm{T}\) underdetermined for only 5 Fano 3folds, which require additional computations by hand (2 of which straightforward). These cases will be treated in Sect. 4.
Remark 3.4
For many deformation families of Fano 3folds one can of course envision alternative methods, e.g. using descriptions as a blowup, double cover or product. We will not discuss the details for these alternative methods as they do not allow for an automated approach. One potential benefit (for certain applications) of these methods could be that they give a more intrinsic description of the cohomology. Let us just point out that they are used for 5 explicit instances in Sect. 4.
Setup and notation. Let us introduce some notation, which is also the notation we use in (the documentation of) the ancillary code. Let X be a Fano 3fold (not of type \({{\textrm{MM}}}_{2.1}\) or \({{\textrm{MM}}}_{2.3}\)), defined by the vanishing of a global section of a vector bundle \({\mathcal {E}}\) inside a key variety F with \({{\,\textrm{codim}\,}}_F X = {{\,\textrm{rank}\,}}{\mathcal {E}}\). By Theorems 3.5 and 3.11 the key variety can be chosen as either a product of Grassmannians or a (possibly singular) toric variety. We wish to compute the cohomology of
We will do this by using the conormal sequence, using that the ideal sheaf \({\mathcal {I}}\) cutting out X gives \(({\mathcal {I}}/{\mathcal {I}}^2)_X\cong {\mathcal {E}}^\vee _X\). Since X is smooth and locally complete intersection within F, one has that \(X \subset F^{\text {sm}}\), hence \(\Omega _F^1_X\) is locally free. From [52, Tags 06AA and 0B3P] it follows that the conormal sequence
is an exact sequence of vector bundles on X. We will twist this sequence by the anticanonical bundle \(\omega _X^\vee \cong \omega _F^\vee _X\otimes \det {\mathcal {E}}_X\). We are interested in computing the cohomologies of the last term of
The first two terms can be resolved by suitable twists of the Koszul complex
The whole point of this reduction is that the tensor product of \(\bigwedge \nolimits ^i{\mathcal {E}}^\vee \) with either of the first two bundles from (28) can now be expressed in terms of vector bundles on F for which good computational methods exist:

for toric varieties we can use the work of Eisenbud–Mustaţă–Stillman [18], as implemented in [50], even when the cotangent sheaf is not locally free by using the reflexive hull of Zariski 1forms;

for homogeneous varieties we can use the Borel–Weil–Bott theorem.
3.3 Complete intersections in toric varieties
The majority of the cases will be covered by this method. The starting point is the following theorem, which follows from the casebycase analysis performed in [15] for Picard ranks \(2,\ldots ,5\), whilst for Picard ranks \(1, 6,\ldots ,10\) it follows from the description using weighted projective spaces and del Pezzo surfaces. The result is valid for every member of the deformation family, which is checked in each section of op. cit. in the paragraph titled “The two constructions coincide”.
Theorem 3.5
(Coates–Corti–Galkin–Kasprzyk) Let X be a Fano 3fold. Assume its deformation family is not of type

Picard rank 1: \({{\textrm{MM}}}_{1.5}\), \({{\textrm{MM}}}_{1.6}\), \({{\textrm{MM}}}_{1.7}\), \({{\textrm{MM}}}_{1.8}\), \({{\textrm{MM}}}_{1.9}\), \({{\textrm{MM}}}_{1.10}\), \({{\textrm{MM}}}_{1.15}\);

Picard rank 2: \({{\textrm{MM}}}_{2.14}\), \({{\textrm{MM}}}_{2.17}\), \({{\textrm{MM}}}_{2.20}\), \({{\textrm{MM}}}_{2.21}\), \({{\textrm{MM}}}_{2.22}\), \({{\textrm{MM}}}_{2.26}\).
Then X has a description as a complete intersection of codimension at most 3 in a projective toric variety F.
Here it is important to point out that

F is singular if the deformation family of X is of type

Picard rank 1: \({{\textrm{MM}}}_{1.1}\), \({{\textrm{MM}}}_{1.11}\), \({{\textrm{MM}}}_{1.12}\);

Picard rank 2: \({{\textrm{MM}}}_{2.1}\), \({{\textrm{MM}}}_{2.2}\), \({{\textrm{MM}}}_{2.3}\), \({{\textrm{MM}}}_{2.8}\);

Picard rank 3: \({{\textrm{MM}}}_{3.1}\), \({{\textrm{MM}}}_{3.4}\), \({{\textrm{MM}}}_{3.14}\), \({{\textrm{MM}}}_{3.16}\);

Picard rank 4: \({{\textrm{MM}}}_{4.5}\);

Picard rank 5: \({{\textrm{MM}}}_{5.1}\);

Picard rank 9: \({{\textrm{MM}}}_{9.1}\);

Picard rank 10: \({{\textrm{MM}}}_{10.1}\);


X is the intersection of Cartier divisors if its deformation family is not of type

Picard rank 2: \({{\textrm{MM}}}_{2.1}\), \({{\textrm{MM}}}_{2.3}\).

So 90 (resp. 92) out of 105 deformation families admit a description in terms of a toric variety F and a vector bundle \({\mathcal {E}}\) (resp. reflexive sheaf) so that we can use the combination of the Koszul sequence and the conormal sequence. We will restrict ourselves to the case where \({\mathcal {E}}\) is a vector bundle, and we will deal with the 2 remaining cases \({{\textrm{MM}}}_{2.1}\), \({{\textrm{MM}}}_{2.3}\) using birational methods in Sect. 4. We remark that it is certainly possible to find suitable models for them as complete intersections of Cartier divisors in different toric varieties, but we did not manage to fully determine the cohomology of \(\bigwedge \nolimits ^2\textrm{T}_X\) in this way.
Remark 3.6
Comparing Theorem 3.5 to what is written in [15] one notices that we write that F is possibly singular, whereas op. cit. seems to require F is smooth for the computation of the quantum period of X, see [15, Assumption D.1]. The argument in the “The two constructions coincide” paragraph of [15] in fact uses a resolution of singularities G of the variety we denote F, where we have obtained F using Magma’s FanWithWeights method to translate the GIT description into a toric description. The discrepancy between G and F is explained in detail for \({{\textrm{MM}}}_{5.1}\) in [55, §11].
Because one can check that \(X\subset G\) is disjoint of the exceptional locus of the small contraction \(G\rightarrow F\) (e.g. by computing the Hodge numbers of X, which show no contraction of X takes place) we can indeed perform the computation for \(X\subset F\) using the sheaf of Zariski 1forms \(\smash {\widehat{\Omega _F^1}}\), for every X in the deformation family.
In Table 1 we give an overview of the codimension of X in F, and whether the computational methods can give a fully determined answer for the cohomology of \(\bigwedge \nolimits ^2\textrm{T}_X\).

The case \({{\textrm{MM}}}_{1.1}\) can be easily determined from the toric computation together with Kodaira vanishing, see Proposition 4.1.

The case \({{\textrm{MM}}}_{4.13}\) can be computed using the description as a blowup, see Proposition 4.6.

The case \({{\textrm{MM}}}_{10.1}\) readily follows from applying the Künneth formula to \({\mathbb {P}}^1\times \textrm{dP}_8\) (Proposition 4.7). For \({{\textrm{MM}}}_{9.1}\) a similar argument using \({\mathbb {P}}^1\times \textrm{dP}_7\) holds, but we chose to use its description as a homogeneous zero locus, see Table 2.
Remark 3.7
The description in [15] describes F as the GIT quotient of an affine space by a torus. To compute cohomology of coherent sheaves on the toric variety F we need to translate this description to a toric fan, and describe the divisors cutting out X in this language. See [15, §C] for some background.
Remark 3.8
It is easy to find examples of deformation families of Fano 3folds in which the cohomology of \(\textrm{T}_X\) is not constant in families, see [47] for a detailed study. A famous example of this is the 6dimensional family \({{\textrm{MM}}}_{1.10}\), members of which are zero loci of \((\bigwedge \nolimits ^2{\mathcal {U}}^\vee )^{\oplus 3}\) on \({{\,\textrm{Gr}\,}}(3,7)\), which generically has finite automorphism group and thus \({{\,\textrm{h}\,}}^0(X,\textrm{T}_X)=0\), but the Mukai–Umemura 3fold has automorphism group \({{\,\textrm{PGL}\,}}_2\) [44, §6] and thus \({{\,\textrm{h}\,}}^0(X,\textrm{T}_X)=3\). An arguably easier example is that of the family \({{\textrm{MM}}}_{2.24}\) [47, §10], members of which are (1, 2)divisors on \({\mathbb {P}}^2\times {\mathbb {P}}^2\). Using coordinates \(x_i\) for the first factor and \(y_i\) for the second, the bihomogeneous equation \(x_0y_0^2+x_1y_1^2+x_2y^2=0\) will give \({{\,\textrm{h}\,}}^0(X,\textrm{T}_X)=2\) whereas the generic X in this family has \({{\,\textrm{h}\,}}^0(X,\textrm{T}_X)=0\).
An example: \({{\textrm{MM}}}_{2.8}\). We now describe an example of a toric complete intersection, and the different steps in the computation. We will consider the deformation family \({{\textrm{MM}}}_{2.8}\), whose Mori–Mukai description is given by

(1)
a double cover of \({{\,\textrm{Bl}\,}}_p{\mathbb {P}}^3\) with anticanonical branch locus B such that \(B\cap E\) is smooth,

(2)
a double cover of \({{\,\textrm{Bl}\,}}_p{\mathbb {P}}^3\) with anticanonical branch locus B such that \(B\cap E\) is singular but reduced,
where E denotes the exceptional divisor of the blowup \({{\,\textrm{Bl}\,}}_p{\mathbb {P}}^3\rightarrow {\mathbb {P}}^3\), and the second is a specialisation of the first.
Proposition 3.9
Let X be a Fano 3fold in the deformation family \({{\textrm{MM}}}_{2.8}\). Then we have that \({{\,\textrm{h}\,}}^i(X,\bigwedge \nolimits ^2\textrm{T}_X)=3,1,1\) for \(i=0,1,2\).
L  1  1  1  \(1\)  0  1 
M  0  0  0  1  1  1 
By [15, §25] the GIT description of the toric key variety F for X is given by the weights so that the nef cone of F is spanned by L and \(L+M\). Then X is a divisor in the linear system \(2L+2M\). Translating this to a description using the set of rays R and the set of cones C gives
The key variety F is now singular, but as explained in Remark 3.6 we can apply the same method as for F smooth.
Proof of Proposition 3.9
We want to compute the cohomology of the first two terms in the sequence (27) twisted by \(\omega _F^\vee (2L2M)\) (which is \(\omega _X^\vee \) before adjunction), so by the Koszul sequence we want to compute the cohomology of the first two terms in the sequences
and
One computes that
for \(i=0,\ldots ,4\), which implies the statement after a diagram chase. \(\square \)
Remark 3.10
The homogeneous description from [16] involves a vector bundle on \({\mathbb {P}}^2\times {\mathbb {P}}^3\times {\mathbb {P}}^{12}\) which is not completely reducible, making the description as a toric complete intersection much more economical.
3.4 Zero loci of sections of homogeneous vector bundles
By combining the description in [15] and [16, Theorems 1.1 and 1.2] for the remaining cases we can state the following.
Theorem 3.11
Let X be a Fano 3fold. Assume its deformation type is not covered by Theorem 3.5, or is \({{\textrm{MM}}}_{9.1}\). Then X is the zero locus of a global section of a completely reducible homogeneous vector bundle on a product of Grassmannians. The description is given in Table 2.
In [16] this is only stated for a generic member for 85 out of 105 deformation families of Fano 3folds. We need it for every member of the deformation family, but only for the 14 cases specified in Theorem 3.11 which are listed in Table 2, which is why we explain now why Theorem 3.11 holds for every member in those families.
Proof
In rank 1 the description is due to Mukai and is classical. Likewise for \({{\textrm{MM}}}_{9.1}\) we just use the classical description of a del Pezzo double plane as a (2, 2)divisor in \({\mathbb {P}}^2\times {\mathbb {P}}^1\).
Let us check the remaining cases. For \({{\textrm{MM}}}_{2.14}\) we take the description as a (1, 1)divisor on \({{\textrm{MM}}}_{1.15}\times {\mathbb {P}}^2\) from [15, §31] (which encodes the blowup) and write the factor \({{\textrm{MM}}}_{1.15}\) using \({{\,\textrm{Gr}\,}}(2,5)\). By loc. cit. every member of the deformation family is realised in this way.
For \({{\textrm{MM}}}_{2.17}\), \({{\textrm{MM}}}_{2.20}\) and \({{\textrm{MM}}}_{2.21}\) the description we use in Table 2 is verbatim that of [15, §34, §37, §38], and by loc. cit. every member of the deformation family can be written as such. For \({{\textrm{MM}}}_{2.22}\) it suffices to observe that [16, Lemma 2.2] applies to every member of the deformation family. Finally, for \({{\textrm{MM}}}_{2.26}\) the proof of the identification in [16, page 676] is given for every member of the deformation family. \(\square \)
An example: \({{\textrm{MM}}}_{2.17}\). In this subsection we exhibit a detailed example of the computation where the Fano 3fold does not admit (at least a priori) a model as a complete intersection in a suitable toric variety. We will use the description given in [15, §34] and [16, Table 1] and recalled in Table 2. The deformation family \({{\textrm{MM}}}_{2.17}\), originally described by Mori and Mukai as the blow up of the quadric 3fold in an elliptic quintic, is realised as the zero locus \({\mathscr {Z}}({\mathcal {E}}) \subset F :={{\,\textrm{Gr}\,}}(2,4) \times {\mathbb {P}}^3\) where
is a rank 4 vector bundle.
Proposition 3.12
Let X be a Fano 3fold in the deformation family \({{\textrm{MM}}}_{2.17}\). Then we have that \({{\,\textrm{h}\,}}^i(X,\bigwedge \nolimits ^2\textrm{T}_X)=5,0,0\) for \(i=0,1,2\).
Proof
We will follow the strategy used in [16, §3.3] and summarised in Sect. 3.2. We need to compute the cohomologies of the first two terms in (28), which are resolved by exact complexes of locally free sheaves, namely the twists of the Koszul complex (29) by \({\mathcal {E}}^\vee \otimes \omega _F^\vee \otimes \det {\mathcal {E}}\) and \(\Omega _F^1 \otimes \omega _F^\vee \otimes \det {\mathcal {E}}\). In this case we have
Each term of the locally free resolutions is a completely reducible vector bundle on F, and we can use the Borel–Weil–Bott theorem to compute its cohomology. It turns out that there are only 2 nonzero cohomology groups for the first two terms of (28) tensored with \(\bigwedge \nolimits ^i{\mathcal {E}}^\vee \) before restriction, for with \(i=0,\ldots ,4={{\,\textrm{rank}\,}}{\mathcal {E}}\), namely
From this we get that the only nonzero cohomologies of the first two terms of (28) are
and the statement follows. \(\square \)
Remark 3.13
It is possible to apply the description as a zero locus in a homogeneous variety to every family of Fano 3folds, but for the purpose of this paper we only do this for the 14 cases listed in Table 2.
The benefit of the toric description is that the codimension is usually (much) lower, making the computation faster and having less places where indeterminacies can occur. E.g. for \({{\textrm{MM}}}_{3.9}\) the description from [16] has codimension 25, which requires a lengthy Koszul computation. Another complication in the computations in the homogeneous setting is that for some Fano 3folds the homogeneous bundle used in the description is not completely reducible.
4 Underdetermined cases
As discussed above there are just a few cases which require additional computations. These are \({{\textrm{MM}}}_{1.1}\), \({{\textrm{MM}}}_{2.1}\), \({{\textrm{MM}}}_{2.3}\), \({{\textrm{MM}}}_{4.13}\), and \({{\textrm{MM}}}_{10.1}\). We will collect the details for them here. For \({{\textrm{MM}}}_{2.1}\), \({{\textrm{MM}}}_{2.3}\) and \({{\textrm{MM}}}_{4.13}\) they are somewhat tedious cohomology computations using the birational description. If it were not for the efficiency of the toric and homogeneous computations the majority of the Fano 3folds would have to be tackled in this way.
The first one is straightforward.
Proposition 4.1
Let X be a Fano 3fold in the deformation family \({{\textrm{MM}}}_{1.1}\). Then we have that \({{\,\textrm{h}\,}}^i(X,\bigwedge \nolimits ^2\textrm{T}_X)=0,0,35\) for \(i=0,1,2\).
Proof
In this case X is a sextic hypersurface in the weighted projective space \({\mathbb {P}}(1^4,3)\). Using the method from Sect. 3.3 on this description for the toric variety \({\mathbb {P}}(1^4,3)\) we immediately obtain that \({{\,\textrm{h}\,}}^i(X,\bigwedge \nolimits ^2\textrm{T}_X)=0,0,35+a,a\) for \(i=0,1,2,3\) for some \(a\ge 0\). But by Lemma 3.1 we have that \({{\,\textrm{h}\,}}^3(X,\bigwedge \nolimits ^2\textrm{T}_X)=0\), so \(a=0\).
Alternatively, one can use that this is a double cover \(f:X\rightarrow Y\) of \({\mathbb {P}}^3=Y\) with a smooth sextic surface S as branch locus. To do so, recall the short exact sequence
and the isomorphism
from [19, §2.3 and Lemma 3.16(d)], together with the isomorphism
This allows one to compute \({{\,\textrm{H}\,}}^i(X,\Omega _X^1\otimes \omega _X^\vee )\), and the only nonvanishing cohomology lives in degree 2 and is isomorphic to
which is 35dimensional. \(\square \)
For the next three cases we will resort to the birational description by Mori–Mukai. Namely we will consider the situation of a Fano 3fold X which is the blowup of a complete intersection curve Z inside another Fano 3fold Y. Let us denote the blowup square as
and consider the short exact sequence
We wish to compute the cohomology of the middle term after twisting by \(\omega _X^\vee =f^*(\omega _Y^\vee )\otimes {\mathcal {O}}_X(E)\), i.e. we will consider the short exact sequence
Lemma 4.2
With the setup from (42) we have that
Proof
Since Z has codimension 2, we have that \(\Omega _{E/Z}^1\cong {\mathcal {O}}_p(2)\otimes p^*({\mathcal {L}})\) for some line bundle \({\mathcal {L}}\) on Z, whilst \(\omega _X_E={\mathcal {O}}_p(1)\). This implies that \(j_*(\Omega _{E/Z}^1)\otimes \omega _X^\vee \cong j_*({\mathcal {O}}_p(1)\otimes p^*{\mathcal {L}})\). But then the vanishing of \({\textbf{R}}p_*{\mathcal {O}}_p(1)\) ensures that
which is what we wanted to show. \(\square \)
Corollary 4.3
With the setup from (42) we have that
Proof
This follows from (44), the vanishing in Lemma 4.2, the isomorphism \({\textbf{R}}f_*({\mathcal {O}}_X(E))\cong {\mathcal {I}}_Z\), and adjunction. \(\square \)
We now consider the underdetermined cases \({{\textrm{MM}}}_{2.1}\) and \({{\textrm{MM}}}_{2.3}\). In this case the methods of Sect. 3.3 don’t necessarily apply: both are described as a codimension2 complete intersection in a singular toric projective variety, and in both cases one of the divisors is not Cartier. Therefore we cannot ensure that the computational (underdetermined) answer is correct,^{Footnote 2} so we will combine Corollary 4.3 with Dolgachev’s computation of sheaf cohomology on weighted projective spaces. We will repeatedly make use of Dolgachev’s formulae for twisted Hodge numbers on weighted projective spaces, cf. [17, §2.3.2\(\)2.3.5].
Proposition 4.4
Let X be a Fano 3fold in the deformation family \({{\textrm{MM}}}_{2.1}\). Then we have that \({{\,\textrm{h}\,}}^i(X,\bigwedge \nolimits ^2\textrm{T}_X)=1,2,7\) for \(i=0,1,2\).
Proof
Such a variety is described as the blowup of a Fano 3fold Y of deformation type \({{\textrm{MM}}}_{1.11}\) in an elliptic curve obtained as complete intersection of two halfanticanonical divisors, and Y is given as a sextic hypersurface in the weighted projective space \({\mathbb {P}}:={\mathbb {P}}(1^3,2,3)\).
Alternatively, X is a (1, 1)section of \(F:=Y\times {\mathbb {P}}^1\). We will first use this description to determine \({{\,\textrm{h}\,}}^0\) and \({{\,\textrm{h}\,}}^2{{\,\textrm{h}\,}}^1\), and then use the blowup description to determine \({{\,\textrm{h}\,}}^2\) (and \({{\,\textrm{h}\,}}^1{{\,\textrm{h}\,}}^0\)), which determines everything.
First step: weighted projective space computation. Consider the conormal sequence for the inclusion \(X\hookrightarrow F\) twisted by the anticanonical line bundle, which by the adjunction formula is
which yields
We claim that
This claim follows from \({{\,\textrm{h}\,}}^0(F,\Omega ^1_F(1,1)_X)=2\) and \({{\,\textrm{h}\,}}^2(F,\Omega ^1_F(1,1)_X){{\,\textrm{h}\,}}^1(F,\Omega ^1_F(1,1)_X)=5\) together with (49), so let us prove this. Consider the twisted Koszul sequence
As \(\Omega ^1_F\) is the direct sum of the pullbacks of the cotangent bundles of Y and \({\mathbb {P}}^1\), one readily gets from the Hodge numbers of \({\mathbb {P}}^1\) and \({{\textrm{MM}}}_{1.11}\) that \({{\,\textrm{h}\,}}^i(F,\Omega ^1_F)=0,2,21,0,0\) for \(i=0,1,2,3,4\). Thus it remains to compute the cohomologies of the middle term of (51). To do that, we apply the Künneth formula to
the second term is acyclic since \({\mathcal {O}}_{{\mathbb {P}}^1}(1)\) is, whilst for the first one we get
To compute the latter, we consider the twisted conormal sequence for Y, seen as a sextic hypersurface in \({\mathbb {P}}\):
By [17, §2.3.2\(\)2.3.5] the first term has only one nonvanishing cohomology \({{\,\textrm{h}\,}}^3(Y,{\mathcal {O}}_Y(5)) = 14\). Thus, we have
The cohomologies of \(\Omega ^1_{{\mathbb {P}}}(1)_Y\) can be obtained as usual by means of a twisted Koszul complex
Now [17, §2.3.2\(\)2.3.5] yields the cohomologies for the first two terms, whence we deduce that
We plug the last equalities into (55) and get
for some integer \(\alpha \ge 0\). By (53) and (51) we get
for some integers \(\alpha ,\beta \ge 0\). But by (49) and Lemma 3.1 we get \(2\alpha ={{\,\textrm{h}\,}}^3(X,\bigwedge \nolimits ^2\textrm{T}_X)=0\), hence the claim.
Second step: blowup computation. We consider the short exact sequence cutting out Z inside Y and tensor it with \(\Omega _Y^1\otimes \omega _Y^\vee \) to obtain
The cohomology of the middle term is determined by the methods in Sect. 3.3 and is given by
To compute the cohomology of the third term, consider the conormal sequence for Z in Y twisted by \(\omega _Y^\vee _Z\cong {\mathcal {O}}_Z(2)\) (because Y is of index 2), which, since Z is an elliptic curve, reads
As Z is cut out by two halfanticanonical divisors, we obtain that \({\mathcal {I}}_Z/{\mathcal {I}}_Z^2\otimes {\mathcal {O}}_Z(2)\cong {\mathcal {O}}_Z(1)^{\oplus 2}\). It now suffices to compute the cohomology of the line bundles \({\mathcal {O}}_Z(1)\) and \({\mathcal {O}}_Z(2)\), which is concentrated in degree 0 by degree reasons, and is determined by a Hilbert series computation on the weighted projective space. We obtain \({{\,\textrm{h}\,}}^i(Z,(\Omega _Y^1\otimes \omega _Y^\vee )_Z)=4,0\) for \(i=0,1\). Combining this with (61) we get that
Together with (50) and Corollary 4.3 this finishes the computation. \(\square \)
Next we tackle the underdetermined case \({{\textrm{MM}}}_{2.3}\) in exactly the same way. In the proof we will explain which details of the computation change.
Proposition 4.5
Let X be a Fano 3fold in the deformation family \({{\textrm{MM}}}_{2.3}\). Then we have that \({{\,\textrm{h}\,}}^i(X,\bigwedge \nolimits ^2\textrm{T}_X)=1,3,1\) for \(i=0,1,2\).
Proof
Such a variety is described as the blowup of a Fano 3fold Y of deformation type \({{\textrm{MM}}}_{1.12}\) in an elliptic curve obtained as complete intersection of two halfanticanonical divisors, and Y is given as a quartic hypersurface in the weighted projective space \({\mathbb {P}}:={\mathbb {P}}(1^4,2)\).
Alternatively, X is a (1, 1)section of \(F=Y\times {\mathbb {P}}^1\). Computing \({{\,\textrm{h}\,}}^i(X,\bigwedge \nolimits ^2\textrm{T}_X) = {{\,\textrm{h}\,}}^i(X,\Omega ^1_X(1,1))\) is analogous to the twostep procedure from Proposition 4.4, so we only summarise the relevant differences in the numerology appearing.
First step: weighted projective space computation (abbreviated). This goes along the same lines as the first step in the proof of Proposition 4.4, except that the claim (50) now becomes
From the Hodge numbers of \({\mathbb {P}}^1\) and \({{\textrm{MM}}}_{1.12}\) we now get \({{\,\textrm{h}\,}}^i(F,\Omega ^1_F)=0,2,10,0,0\) for \(i=0,1,2,3,4\). The twisted conormal sequence for Y, which is a quartic hypersurface in \({\mathbb {P}}\), gives \({{\,\textrm{h}\,}}^3(Y,{\mathcal {O}}_Y(3)) = 4\). We obtain
for some integer \(\alpha \ge 0\). As in the proof of Proposition 4.4 we get
for some integers \(\alpha , \beta \ge 0\). But by Lemma 3.1 we get \(2\alpha ={{\,\textrm{h}\,}}^3(X,\bigwedge \nolimits ^2\textrm{T}_X)=0\), hence the claim.
Second step: blowup computation. This goes along the same lines as the second step in the proof of Proposition 4.4, except that (61) now becomes
and that the Hilbert series computation is performed for an elliptic curve in the weighted projective space \({\mathbb {P}}(1^4,2)\). We obtain \({{\,\textrm{h}\,}}^i(Z,(\Omega _Y^1\otimes \omega _Y^\vee )_Z)=8,0\) for \(i=0,1\). Combining this with (67) we get that
Together with (64) and Corollary 4.3 this finishes the computation. \(\square \)
The next underdetermined case is the Fano 3fold which was originally missed in the classification [42]. The method is similar to what we did for \({{\textrm{MM}}}_{2.1}\) and \({{\textrm{MM}}}_{2.3}\) using the birational description, but requires less work.
Proposition 4.6
Let X be a Fano 3fold in the deformation family \({{\textrm{MM}}}_{4.13}\). Then we have that \({{\,\textrm{h}\,}}^i(X,\bigwedge \nolimits ^2\textrm{T}_X)=4,0,0\) for \(i=0,1,2\).
Proof
Such a variety is described as the blowup of the Fano 3fold \(Y={\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1\) of deformation type \({{\textrm{MM}}}_{3.27}\) in a rational curve Z of tridegree (1, 1, 3). The curve Z is given as a complete intersection of type (2, 1, 1) and (1, 1, 0), see [15, §86].
By Corollary 4.3 we want to compute \({{\,\textrm{H}\,}}^\bullet (Y,\Omega _Y^1\otimes \omega _Y^\vee \otimes {\mathcal {I}}_Z)\). Denote by \(p_{i,j}:Y\rightarrow {\mathbb {P}}^1\times {\mathbb {P}}^1\) the three projections. Using the isomorphism
and the projection formula we want to compute the cohomology of
But \(p_{i,j,*}{\mathcal {I}}_Z={\mathcal {I}}_{Z_{i,j}}\) where \(Z_{i,j}\) is the image of Z under \(p_{i,j}\), which is in turn a divisor of bidegree (1, 1), resp. (1, 3) and (1, 3) on \({\mathbb {P}}^1\times {\mathbb {P}}^1\). This reduces the computation to the cohomology of \({\mathcal {O}}_{{\mathbb {P}}^1\times {\mathbb {P}}^1}(1,1)\) and \({\mathcal {O}}_{{\mathbb {P}}^1\times {\mathbb {P}}^1}(1,1)^{\oplus 2}\), where the latter is cohomologyfree and the former has cohomology concentrated in degree 0, where it is 4dimensional. \(\square \)
The final case is the product of a del Pezzo surface of degree 1 with \({\mathbb {P}}^1\), and the computation is immediate (e.g. using the description of Appendix B and the Künneth formula).
Proposition 4.7
Let X be a Fano 3fold in the deformation family \({{\textrm{MM}}}_{10.1}\). Then we have that \({{\,\textrm{h}\,}}^i(X,\bigwedge \nolimits ^2\textrm{T}_X)=2,24,0\) for \(i=0,1,2\).
Notes
i.e. cannot be written as the blowup in a curve of a Fano 3fold of lower Picard rank.
The underdetermined result from the toric computation is nevertheless consistent with the final answer, so a more detailed analysis of the toric description might be valid. But the indeterminacy needs to be dealt with by alternative methods in any case.
References
Artenstein, Dalia, Lanzilotta, Marcelo, Solotar, Andrea: Gerstenhaber structure on Hochschild cohomology of toupie algebras. Algebr. Represent. Theory 23(2), 421–456 (2020). https://doi.org/10.1007/s1046801909854y
Bartocci, Claudio, Macrì, Emanuele: Classification of Poisson surfaces. Commun. Contemp. Math. 7(1), 89–95 (2005). https://doi.org/10.1142/S0219199705001647
Belmans, Pieter: Fanography. https://fanography.info
Belmans, Pieter: Hochschild cohomology of noncommutative planes and quadrics. J. Noncommut. Geom. 13(2), 769–795 (2019). https://doi.org/10.4171/JNCG/338
Belmans, Pieter, Fatighenti, Enrico, Tanturri, Fabio: Bivector fields for Fano 3folds. https://github.com/pbelmans/bivectorfieldsfano3folds
Belmans, Pieter, Fu, Lie, Raedschelders, Theo: Hilbert squares: derived categories and deformations. Selecta Math. (N.S.) 25(3), Paper No. 37, 32 (2019). https://doi.org/10.1007/s000290190482y
Belmans, Pieter, Smirnov, Maxim: Hochschild cohomology of generalised Grassmannians. accepted for publication in Documenta Mathematica arXiv:1911.09414v1 [math.AG]
Bernardara, Marcello, Fatighenti, Enrico, Manivel, Laurent, Tanturri, Fabio: Fano fourfolds of K3 type (2021). arXiv:2111.13030 [math.AG]
Bondal, Alexey: Noncommutative deformations and Poisson brackets on projective spaces. eprint: MPI/9367. https://www.mpimbonn.mpg.de/preblob/4912
Bosma, Wieb, Cannon, John, Playoust, Catherine: The Magma algebra system. The user language. I. J. Symbolic Comput. 24(3–4), 235–265 (1997). https://doi.org/10.1006/jsco.1996.0125. (Computational algebra and number theory (London, 1993). 07477171)
Calaque, Damien, Van den Bergh, Michel: Hochschild cohomology and Atiyah classes. Adv. Math. 224(5), 1839–1889 (2010). https://doi.org/10.1016/j.aim.2010.01.012
Cǎldǎraru, Andrei: The Mukai pairing. II. The HochschildKostantRosenberg isomorphism. Adv. Math. 194(1), 34–66 (2005). https://doi.org/10.1016/j.aim.2004.05.012
Cheltsov, Ivan, Przyjalkowski, Victor: KatzarkovKontsevichPantev Conjecture for Fano threefolds. arXiv:1809.09218v1 [math.AG]
Chen, Yuan, Guo, Yanhong, Yunge, Xu.: The Gerstenhaber bracket of Hochschild cohomology of triangular quadratic monomial algebra. Indian J. Pure Appl. Math. 46(2), 175–190 (2015). https://doi.org/10.1007/s1322601501200
Coates, Tom, Corti, Alessio, Galkin, Sergey, Kasprzyk, Alexander: Quantum periods for 3dimensional Fano manifolds. Geom. Topol. 20(1), 103–256 (2016). https://doi.org/10.2140/gt.2016.20.103
Biase, De.: Lorenzo, Fatighenti, Enrico, Tanturri, Fabio: Fano 3folds from homogeneous vector bundles over Grassmannians. Rev. Mat. Complut. 35(3), 649–710 (2022). https://doi.org/10.1007/s13163021004012. (11391138)
Dolgachev, Igor: Weighted projective varieties. In: Group actions and vector fields (Vancouver, B.C., 1981). vol. 956. Lecture Notes in Math. pp. 34–71. Springer, Berlin (1982). https://doi.org/10.1007/BFb0101508
Eisenbud, David, Mustaţǎ, Mircea, Stillman, Mike: Cohomology on toric varieties and local cohomology with monomial supports. J. Symbolic Comput. 29(4–5), 583–600 (2000). https://doi.org/10.1006/jsco.1999.0326. (Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998))
Esnault, Hélène., Viehweg, Eckart: Lectures on vanishing theorems. DMV Seminar, vol. 20, p. vi+164. Birkhäuser Verlag, Basel (1992). https://doi.org/10.1007/9783034886000 . (isbn: 3764328223)
Grayson, Daniel R., Stillman, Michael E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
Hemelsoet, Nicolas, Voorhaar, Rik: A computer algorithm for the BGG resolution. J. Algebra 569, 758–783 (2021). https://doi.org/10.1016/j.jalgebra.2020.09.043
Hong, Wei: Holomorphic polyvector fields on toric varieties. arXiv:2010.07053v1 [math.AG]
Huybrechts, Daniel, Rennemo, Jørgen Vold.: Hochschild cohomology versus the Jacobian ring and the Torelli theorem for cubic fourfolds. Algebr. Geom. 6(1), 76–99 (2019). https://doi.org/10.14231/AG2019005
Iskovskih, Vasiliĭ Alekseevich: Fano threefolds. I. Izv. Akad. Nauk SSSR Ser. Mat. 41(3), 516–562, 717 (1977)
Iskovskih, Vasiliĭ Alekseevich.: Fano threefolds. II. Izv. Akad. Nauk SSSR Ser. Mat. 42(3), 506–549 (1978)
Iskovskih, Vasiliĭ Alekseevich., Prokhorov, Yuri: Fano varieties. Algebraic geometry. Encyclopaedia Math. Sci., vol. 47, pp. 1–247. Springer, Berlin (1999)
Jahnke, Priska, Radloff, Ivo: Fano threefolds with sections in \(\Omega ^1_{V} (1)\). Math. Nachr. 280(1–2), 127–139 (2007). https://doi.org/10.1002/mana.200410469
Katzarkov, Ludmil, Kontsevich, Maxim, Pantev, Tony: BogomolovTianTodorov theorems for LandauGinzburg models. J. Differential Geom. 105(1), 55–117 (2017)
Keller, Bernhard: Derived invariance of higher structures on the Hochschild complex (2005). https://webusers.imjprg.fr/~bernhard.keller/publ/dih.pdf
Kontsevich, Maxim: Deformation quantization of algebraic varieties. Lett. Math. Phys. 56(3), 271–294 (2001). https://doi.org/10.1023/A:1017957408559. (EuroConférence Moshé Flato 2000, Part III (Dijon))
Kontsevich, Maxim: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003). https://doi.org/10.1023/B:MATH.0000027508.00421.bf
Kuznetsov, Alexander: Height of exceptional collections and Hochschild cohomology of quasiphantom categories. J. Reine Angew. Math. 708, 213–243 (2015). https://doi.org/10.1515/crelle20130077
Kuznetsov, Alexander: Hochschild homology and semiorthogonal decompositions. arXiv:0904.4330v1 [math.AG]
Kuznetsov, Alexander, Prokhorov, Yuri, Shramov, Constantin: Hilbert schemes of lines and conics and automorphism groups of Fano threefolds. Jpn. J. Math. 13(1), 109–185 (2018). https://doi.org/10.1007/s1153701717146
Loray, Frank, Pereira, Jorge Vitório, Touzet, Frédéric.: Foliations with trivial canonical bundle on Fano 3folds. Math. Nachr. 286(8–9), 921–940 (2013). https://doi.org/10.1002/mana.201100354
Lowen, Wendy, Van den Bergh, Michel: Deformation theory of abelian categories. Trans. Amer. Math. Soc. 358(12), 5441–5483 (2006). https://doi.org/10.1090/S0002994706038712
Lowen, Wendy, Van den Bergh, Michel: Hochschild cohomology of abelian categories and ringed spaces. Adv. Math. 198(1), 172–221 (2005). https://doi.org/10.1016/j.aim.2004.11.010
Lunts, Valery, Przyjalkowski, Victor: LandauGinzburg Hodge numbers for mirrors of del Pezzo surfaces. Adv. Math. 329, 189–216 (2018). https://doi.org/10.1016/j.aim.2018.02.024
Markarian, Nikita: The Atiyah class, Hochschild cohomology and the RiemannRoch theorem. J. Lond. Math. Soc. (2) 79(1), 129–143 (2009). https://doi.org/10.1112/jlms/jdn064
Mori, Shigefumi, Mukai, Shigeru: Classification of Fano 3folds with \(B_2\ge 2\). Manuscripta Math. 36(2), 147–162 (1981/82). https://doi.org/10.1007/BF01170131
Mori, Shigefumi, Mukai, Shigeru: Classification of Fano 3folds with \(B_2\ge 2\) I. Algebraic and topological theories (Kinosaki, 1984), pp. 496–545. Kinokuniya, Tokyo (1986)
Mori, Shigefumi, Mukai, Shigeru: Erratum: “Classification of Fano 3folds with \(B_2\ge 2\) [Manuscripta Math. 36 (1981/82), no. 2, 147–162; MR0641971 (83f:14032)]’’. Manuscripta Math. 110(3), 407 (2003). https://doi.org/10.1007/s0022900203362
Mukai, Shigeru: Biregular classification of Fano 3folds and Fano manifolds of coindex 3. Proc. Nat. Acad. Sci. U.S.A. 86(9), 3000–3002 (1989). https://doi.org/10.1073/pnas.86.9.3000
Mukai, Shigeru, Umemura, Hiroshi: Minimal rational threefolds. In: Algebraic geometry (Tokyo/Kyoto, 1982). vol. 1016. Lecture Notes in Math. pp. 490–518. Springer, Berlin (1983). https://doi.org/10.1007/BFb0099976
Mustaţǎ, Mircea: Vanishing theorems on toric varieties. Tohoku Math. J. (2) 54(3), 451–470 (2002)
Polishchuk, Alexander: Algebraic geometry of Poisson brackets. J. Math. Sci. (New York) 84(5). Algebraic geometry 7, 1413–1444 (1997). https://doi.org/10.1007/BF02399197
Przhiyalkovskiǐ, Victor, Chel’tsov, Ivan, Shramov, Constantin A.: Fano threefolds with infinite automorphism groups. Izv. Ross. Akad. Nauk Ser. Mat. 83(4), 226–280 (2019). https://doi.org/10.4213/im8834
Pym, Brent: Constructions and classifications of projective Poisson varieties. Lett. Math. Phys. 108(3), 573–632 (2018). https://doi.org/10.1007/s1100501709845
Redondo, María Julia., Roman, Lucrecia: Gerstenhaber algebra structure on the Hochschild cohomology of quadratic string algebras. Algebr. Represent. Theory 21(1), 61–86 (2018). https://doi.org/10.1007/s1046801797041
Smith, Gregory G.: NormalToricVarieties: a package for working with normal toric varieties. Version 1.8. A Macaulay2 package available at https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages
Solotar, Andrea: The Gerstenhaber bracket in Hochschild cohomology: Methods and examples. In: Representation theory and beyond. vol. 758. Contemp. Math. Amer. Math. Soc., pp. 287–298. Providence, RI (2020). https://doi.org/10.1090/conm/758/15240
The Stacks project authors (2021) The Stacks project. https://stacks.math.columbia.edu
Swan, Richard G.: Hochschild cohomology of quasiprojective schemes. J. Pure Appl. Algebra 110(1), 57–80 (1996). https://doi.org/10.1016/00224049(95)000917. (00224049)
Toda, Yukinobu: Deformations and FourierMukai transforms. J. Differential Geom. 81(1), 197–224 (2009)
Totaro, Burt: Bott vanishing for Fano 3folds (2023) . arXiv:2302.08142 [math.AG]
Yekutieli, Amnon: The continuous Hochschild cochain complex of a scheme. Canad. J. Math. 54(6), 1319–1337 (2002). https://doi.org/10.4153/CJM20020518
Acknowledgements
We would like to thank Marcello Bernardara, Alexander Kasprzyk, Brent Pym and Helge Ruddat for interesting conversations. We want to thank Burt Totaro for pointing out the subtletly explained in Remark 3.6. We want to thank the referee for a careful reading and interesting comments that improved the paper. And most of all we want to thank Alexander Kuznetsov for many conversations over the years about Fano 3folds, and comments on an earlier version of this paper. The first author was partially supported by the FWO (Research Foundation–Flanders). The second and third author are members of INdAMGNSAGA.
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Appendices
Appendix A: Dimensions for Fano 3folds
In this appendix we have collected all the polyvector parallelograms of Fano 3folds. The notation we use for this is introduced in (10): writing \({{\,\textrm{p}\,}}^{p,q}:=\dim _k{{\,\textrm{H}\,}}^p(X,\bigwedge \nolimits ^q\textrm{T}_X)\), and using that X is a Fano 3fold, we can summarise the Hochschild–Kostant–Rosenberg decomposition of Hochschild cohomology as
In the case of jumping of \({{\,\textrm{h}\,}}^i(X,\textrm{T}_X)\), we have given the lowest value, and indicated with a * next to the value \({{\,\textrm{h}\,}}^0(X,\textrm{T}_X)\) that jumping occurs.
This information (and more) is also available in a more interactive way at [3].
Rank 1.
Rank 2.
Rank 3.
Rank 4.
Rank 5.
Rank 6.
Rank 7.
Rank 8.
Rank 9.
Rank 10.
Appendix B: Dimensions for del Pezzo surfaces
For ease of reference and completeness’ sake we give the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of del Pezzo surfaces. We use the notation introduced in (10), suitably modified for surfaces.
One can compute this in many ways, e.g. using the methods from Sect. 3.1. There is no cohomology jumping in this case. What is interesting to remark is that one does observe cohomology jumping when extending to noncommutative del Pezzo surfaces, as in [4].
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Belmans, P., Fatighenti, E. & Tanturri, F. Polyvector fields for Fano 3folds. Math. Z. 304, 12 (2023). https://doi.org/10.1007/s00209023032612
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DOI: https://doi.org/10.1007/s00209023032612